Index reduction for rectangular descriptor systems via feedbacks

Vikas Kumar Mishra,Mahendra Kumar Gupta,Nutan Kumar Tomar,James Lam

doi:10.1080/23311916.2017.1319786

Vikas Kumar Mishra, Mahendra Kumar Gupta + Show 2 more

Open Access

https://doi.org/10.1080/23311916.2017.1319786

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Journal: Cogent Engineering	Publication Date: Jan 1, 2017
Citations: 4	License type: CC BY 4.0

Affiliation: Indian Institute of Technology Patna

Abstract

Index plays a fundamental role in the study of descriptor systems. For regular descriptor systems, calculation of the index can be performed by calculating the index of the nilpotent matrix obtained by means of the Weierstrass canonical form. Notwithstanding, if the system is not regular, there is no algebraic technique to determine the index of the system. A sufficient algebraic criterion is provided to determine the index of a general linear time-invariant descriptor systems. Thereafter, we provide an alternate but lucid proof of the fact that impulse controllability is necessary and sufficient for the existence of a semistate feedback such that the closed loop system is of the index at most one. Finally, a sufficient test for the existence of a semistate feedback such that the closed loop system is of the index at most two is provided. Examples are given to illustrate the presented theory.

Highlights

Consider an linear time-invariant (LTI) continuous-time descriptor systemEẋ (t) = Ax(t) + Bu(t), (1)where E, A ∈ Rm×n, and B ∈ Rm×r are known constant matrices
The theorem proves that I-controllability is necessary and sufficient for the existence of a semistate feedback such that the closed loop system is of index at most one
Concluding remarks The paper was devoted to the index determination and reduction for general LTI descriptor systems

Summary

PUBLIC INTEREST STATEMENT

Descriptor systems play a vital role in the modeling of physical systems where dynamics of the system is constrained. It is well known that the solvability of descriptor systems requires certain smoothness assumptions on its input. Loosely speaking, is a measure of the number of differentiations required by the input so that the system has a solution. As higher index problems are difficult to handle, at least, numerically. It is always desirable to minimize the index of the system. There are two known techniques to reduce the index: with feedback and without feedback. We have developed some techniques to determine the index and to reduce the index by applying feedback

Introduction

This implies that for